Diffractive Vector Photoproduction using Holographic QCD

Diffractive Vector Photoproduction using Holographic QCD

Abstract

We discuss diffractive photon-production of vector mesons in holographic QCD. At large , the QCD scattering amplitudes are reduced to the scattering of pair of dipoles exchanging a closed string or a pomeron. We use the holographic construction in AdS to describe both the intrinsic dipole distribution in each hadron, and the pomeron exchange. Our results for the heavy meson photon-production are made explicit and compared to some existing experiments.

pacs:
12.39.Jh, 12.39.Hg, 13.30.Eg
\definechangesauthor

[name=Chang Hwan, color=blue]CHL \definechangesauthor[name=Hui Young, color=orange]RL \definechangesauthor[name=Ismail Zahed, color=purple]IZ

I introduction

Diffractive scattering at high energy is dominated by pomeron exchange, an effective object corresponding to the highest Regge trajectory. The slowly rising cross sections are described by the soft Pomeron with a small intercep (0.08) and vacuum quantum numbers. Reggeon exchanges have even smaller intercepts and are therefore subleading. Reggeon theory for hadron-hadron scattering with large rapidity intervals provide an effective explanation for the transverse growth of the cross sections Gribov:1984tu (). In QCD at weak coupling the pomeron is described through resummed BFKL ladders resulting in a large intercept and zero slope Kuraev:1977fs (); Balitsky:1978ic ().

The soft Pomeron kinematics suggests an altogether non-perturbative approach. Through duality arguments, Veneziano suggested long ago that the soft Pomeron is a closed string exchange Veneziano:1968yb (). In QCD the closed string world-sheet can be thought as the surface spanned by planar gluon diagrams. The quantum theory of planar diagrams in supersymmetric gauge theories is tractable in the double limit of a large number of colors and  t Hooft coupling using the AdS/CFT holographic approach HOLOXX ().

In the past decade there have been several attempts at describing the soft pomeron using holographic QCD Rho:1999jm (); Janik:2000aj (); Brower:2006ea (); Stoffers:2012zw (); Stoffers:2012ai (); Basar:2012jb (). In this paper we follow the work in Stoffers:2012ai () and describe diffractive production through the exchange of a soft pomeron in curved AdS geometry with a soft or hard wall. This is inherently a bottom-up approach HOLOXXX () with the holographic or 5th direction playing the role of the scale dimension for the closed string, interpolating between two fixed size dipoles. We follow the suggestion in Polchinski:2001tt (); Brodsky:2014yha () and describe the intrinsic dipole size distribution of hadrons on the light cone through holographic wave functions in curved AdS. Diffractive production of vector mesons was investigated in the non-holographic context by many in MANY (). Recently a holographic description was explored in Ahmady:2016ujw () in the context of the color glass condensate, and reggeized gravitons in DJURIC ().

The organization of the paper is as follows: In section 2 we briefly review the set up for diffractive scattering through a holographic pomeron as a closed surface exchange in curved AdS with a (hard) wall. In section 3, we detail the construction of the light cone wavefunctions including their intrinsic light cone dipole distributions. In section 4 and 5 we make explicit the AdS model with a (soft) wall to descrive the intrinsic dipole distributions of massive vector mesons. As a check on the intrinsic wavefunctions, we calculate the pertinent vector electromagnetic decay constants. Our numerical results for the partial cross sections and their comparison to vector photoproduction data are given in section 6. Our conclusions are summarized in section 7.

Ii Dipole-Dipole Scattering

Figure 1: Dipole-Dipole Scattering.

In this section we briefly review the set-up for dipole-dipole scattering using an effective string theory. For that we follow Basar:2012jb () and consider the elastic scattering of two dipoles

(1)

as depicted in Fig. 1. is the impact parameter and the relative angle is the Euclidean analogue of the rapidity interval Shuryak:2000df (); Meggiolaro:1997mw ()

(2)

with .

ii.1 Dipole-dipole correlator

Following standard arguments as in Basar:2012jb (), the scattering amplitude in Euclidean space is given by

(3)

with the connected correlator of two Wilson loops, each represented by a rectangular loop sustained by a dipole and slated at a relative angle in Euclidean space as shown in Fig. 1. The leading contribution from a closed string exchange is

(4)

where

(5)

is the string partition function on the cylinder topology with modulus . The sum is over the string world-sheet with specific gauge fixing or ghost contribution. Here is the string coupling.

ii.2 Holographic Pomeron

In flat dimensions, the effective string description for long strings is the Polyakov-Luscher action with . However, the dipole sources for the incoming Wilson loops vary in size within a hadron. To account for this change and enforces conformality at short distances, we follow Stoffers:2012zw () and identify the dipole size with the holographic direction. The stringy exchange in (4) is now in curved AdS in with . At large relative rapidity this exchange is dominated by the string tachyon mode with the result Stoffers:2012zw ()

(6)

and

(7)

refers to the tachyon propagator in walled . It solves a curved diffusion equation in the metric defined by

(8)

within with a zero current at the wall,

(9)

with the chordal distances given by

(10)

and and . The holographic Pomeron intercept and diffusion constant are respectively given by

D (11)

The string coupling in walled AdS is identified as and . Here is an overall dimensionless parameter that takes into account the arbitrariness in the normalization of the integration measure in (4). This analysis of the holographic Pomeron is different from the (distorted) spin-2 graviton exchange in Brower:2006ea () as the graviton is massive in walled AdS. Our approach is similar to the one followed in Basar:2012jb () with the difference that and not 10 Stoffers:2012zw (). It is an effective approach along the bottom-up scenario of AdS. Modulo different parameters, the holographic Pomeron yields a dipole-dipole total cross section that is similar to the one following from BFKL exchanges Mueller:1994gb (); Salam:1995zd (), and a wee-dipole density that is consistent with saturation at HERA GolecBiernat:1998js ().

Iii Photon-hadron scattering

In a valence quark picture an incoming meson is considered as a dipole made of a pair, while a baryon is considered as a dipole made of a pair of a quark-diquark. The quantum scattering amplitude follows by assigning to the scattering pairs dipole sizes and distributing them within the quantum mechanical amplitude of the pertinent hadron. At large the scattering particles propagate along the light cone and are conveniently described by light cone wave functions. Typically, the latters are given in terms of an intrinsic wavefunction for a dipole of size with a fraction of parton longitudinal momentum . With this in mind, the scattering amplitude for the diffractive process for vector meson photo-production , reads

The normalization conforms with the light cone rules.

Note that in flat -space (also for ), the propagator (9) simplifies

after the substitution . For an estimate of (III) we may insert (III) into (III), ignore the wall and assume to carry out the integration in (III) exactly

(15)

with the Pomeron trajectory

(16)

iii.1 Photon wave function

The description of the light cone photon wave function in terms of a pair follows from light cone perturbation theory as described in Lepage:1980fj (). Let be the virtuality of the photon of polarization . The amplitude for finding a pair in the virtual photon with light cone momentum fractions is given by MANY (); Lepage:1980fj ()

(17)

with the matrix entries in helicity of in (III). Here is the charge of a quark of flavor , , and are modified Bessel functions. Also are the 2-dimensional dipole polar coordinates. While the photo-production analysis to be detailed below corresponds to , we will carry the analysis for general for future reference.

iii.2 Hadron wave functions

We start by defining the proton (squared) wave function for a pair of quark-diquark as

(18)

by simply assuming equal sharing of the longitudinal momentum among the pair, and a fixed dipole size , with the normalization

(19)

The vector meson wave function on the light cone will be sought by analogy with the photon wave function given above. Specifically we write

(20)

where are the matrix entries in helicity of in (III). The intrinsic dipole distributions for the vector mesons will be sought below in the holographic construction by identifying the holographic direction in the description of massive vector mesons with the dipole size Polchinski:2001tt (); Brodsky:2014yha ().

iii.3 Partial cross sections

The partial diffractive cross sections for the production of longitudinal and transverse vector mesons are given by

(21)

with the virtual-photon-vector-meson transition amplitudes following from the contraction of the helicity matrix elements (III.1-III.2). The results are

The vector charge is computed as the average charge

in a state with flavor content . The elastic differential cross section follows as

(24)

Iv from holography

The intrinsic light cone distributions in the vector mesons is inherently non-perturbative. Our holographic set-up for the description of the process as a dipole-dipole scattering through a holographic pomeron in AdS suggests that we identify the intrinsic light cone distributions with the holographic wave function of massive Spin-1 mesons in AdS. The mass will be set through a tachyon field in bulk.

iv.1 AdS model for Spin-1

With this in mind, consider an AdS geometry with a vector gauge field and a dimensionless tachyon field described by the non-anomalous action

(25)

with and , and signature The coupling is fixed by standard arguments HOLOXXX (). The background tachyon field satisfies

(26)

which is solved by

(27)

The constants in (27) are fixed by the holographic dictionary HOLOXX (); HOLOXXX () near the UV boundary ()

(28)

In the heavy quark limit , so .

In the presence of , the vector gauge field satisfies

(29)

We now seek a plane-wave vector meson with 4-dimensional spatial polarization in the form

(30)

which yields

(31)

We now use the solution for with (no heavy chiral condensate), and identify with the (constituent) quark mass. Thus near the boundary

(32)

We can now either solve (32) using a hard-wall by restricting (32) to the slab geometry , or introducing a soft wall SOFT (). The former is a Bessel function with a spectrum that does not Reggeizes, while the latter is usually the one favored by the light-cone with a spectrum that Reggeizes. The minimal soft wall amounts

(33)

Defining , it follows that

(34)

The meson spectrum Reggeizes. The value for is fixed by the string tension.

iv.2 Intrinsic wave functions

We now suggest that the holographic wavefunction

(35)

can be related to the intrinsic amplitudes for the dipole distribution in the light cone wavefunctions for the vector mesons in (III.2). For that we note that the main part of the transverse vector in (III.2) satisfies . With this in mind, we identify the holographic coordinate with the relative dipole size through  Polchinski:2001tt (); Brodsky:2014yha (), and match the r-probability of the intrinsic state to the z-probability of the spin-1 state in bulk AdS,

(36)

The extra in the bracket is the warping factor. Solving for we obtain

(37)

which normalizes to 1

(38)

For a massive spin-1 meson with the helicity content and quark mass analogous to the content as ansatz in (37), we will assume the holographic dipole content derived in (37), with instead general overall constants

(39)

are now fixed by the helicity-dependent normalizations using (III.2), i.e.

(40)

More specifically we have

(41)

which fix

(42)

(39) is in agreement with the intrinsic dipole wave function developed in Brodsky:2014yha () using the light cone holographic procedure for . We note that (35) describes a massive spin-1 gauge field in AdS.

V Leptonic Decay constants

The size of the light cone wavefunction is empirically constrained by the electromagnetic decay width as captured by the measured vector decay constant for each of the vector mesons,

(43)

This puts an empirical constraint on the longitudinal and transverse light cone wavefunctions (37) using the holographic intrinsic wavefunctions (39) as suggested earlier.

v.1 Longitudinal

More specifically, the longitudinal wavefunction gives for the right-hand-side in (43)

(44)

as , with the conventions . The left-hand-side in (43) can be reduced using the light cone rules in the Appendix of Lepage:1980fj () together with the longitudinal wavefunction (III.2) to have

(45)

The first bracket refers to the reduction of the current, and the second bracket to the reduction of the longitudinal wavefunction. The result for the vector decay constant from the longitudinal current is

(46)

after the use of the normalization as given in (42). For example, for the rho meson , while for the phi meson .

v.2 Transverse

For a consistency check, the same rules apply to the transverse component of the current . The transverse wavefunction gives for the right-hand side of (43)

(47)

The left-hand-side can be reduced using also the light cone rules

(48)

The first contribution stems from the reduction of the current and the second contribution from the reduction of the transverse wavefunction. The signs in (V.2) follows the assignments. Using the explicit form of the wavefunction (37) and performing an integration by parts, we have the identity

Inserting (V.2) in (V.2) gives for the left-hand-side

(50)

which reduces to

(51)

Substituting the value of from the Regge spectrum (IV.1) yields the transverse to longitudinal ratio for the decay constants

(52)

with . In Fig. 2 we show the behavior of (52) in the range from the massless to the heavy quark limit where it reaches 1.

Figure 2: Ratio (52) of the transverse to longitudinal decay constants vs from the massless to the heavy quark limit.
[GeV] [MeV] [MeV] [GeV]
1000 186.9 1.087 0.325 0.63
(775.3) (204)
971 44.8 1.114 0.302 1.46
(782.7) (59)
1172 78.67 1.096 0.375 0.44
(1020) (74)
3185 153.3 1.344 0.366 1.50
(3097) (90)
9472 49.0 1.367 4.730 0.234 5.00
(9460) (25)
0.1 GeV 3
1.8 GeV 23
Table 1: Holographic parameters along with the model prediction for . See text.

Vi Numerical analysis

To carry out the numerical analysis, we can partially eliminate the model dependence in the tansition amplitudes (LABEL:GV), by trading in the normalizations in (42) with (46) to obtain

Here we have set

(54)

with fixed by the ground state meson mass in (IV.1)

(55)

With the exception of , all holographic parameters are fixed by the DIS analysis in Stoffers:2012zw () as listed in Table 1. For the light vector mesons, we have set at their constituent values, and at their PDG values. The value of is adjusted to reproduce the best value for the vector meson decay constants. The vector masses are then fixed by (55) as listed in Table 1. In our holographic set up, the lower decay constants for the heavier mesons imply smaller values of (string tension) for in comparison to the for instance. Since is a measure of the compactness of the wavefunction at the origin this is reasonable, although the spread in the transverse direction appears to be larger in the absence of the Coulombic interactions which are important for . Finally, the string coupling is adjusted to reproduce the overall normalization of the cross section for each vector meson channel.

vi.1 Radiative widths

In terms of (46), the radiative decay width is

(56)

We note that (46) is finite in the heavy quark limit as expected from the Isgur-Wise symmetry. Using (IV.1), (56) gives

(57)

The emprical ratios of the width to the squared charge are

(58)

with fixed by (III.3). The holographic decay widths are in agreement with the empirical ones for the light vector mesons , but substantially smaller for the heavy vector mesons . This maybe an indication of the strong Coulomb corrections in the heavy quarkonia missing in our current holographic construction. One way to remedy this is through the use of improved holographic QCD KIRITSIS ().

vi.2

In Fig. 3 we show the differential -photoproduction versus for GeV. At this energy the photon size is of the order of the hadronic sizes and sensitive to non-perturbative physics. In Fig. 4 we show the total cross section for -photoproduction in the range of low mass photons. The discrepancy close to treshold maybe due to t-channel sigma-exchange and the s-channel photo-excitation of the in the intermediate nucleon state, not retained in our analysis. Note that both the and have comparable transverse sizes with but very different decay constants. We expect their differential and total cross sections to be in the ratio of their decay constant, say .

Figure 3: Differential cross section for versus for : the solid line is this work, the filled circle are the data. Data are taken from Ballam:1971yd ().
Figure 4: Total cross section for versus . The solid line is this work, the filled circles are the data. Data are taken from Barth:2003kv ().

vi.3

In Figs. 5-8, we present the total and differential cross section for the -photoproduction process. In Fig. 6 we compare our results to the available CLAS and LEPS data. Our results agree with the backward angle data well, but overshoot the forward angle data. In Fig. 7-8, the differential cross sections are shown. The agreement at large probes mostly the Pomeron exchange. Note that our overall fit to the -decay constant implies a transverse size for the that is comparable to the sizes, which is reasonable. The differential and total cross sections are expected to be in the ratio of the squared decay constants or .

Figure 5: Total cross section for from threshold to GeV. Data are taken from Ballam:1972eq (); Derrick:1996af (); Busenitz:1989gq ().
Figure 6: Differential cross section for in low energy region. The black solid line is the present work. Data at and 2.365 GeV (red up triangle) are taken from the charged mode Dey:2014tfa () and at and 2.38 GeV (blue down triangle) from the neutral mode Seraydaryan:2013ija () in the CLAS Collaboration.
Figure 7: Differential cross section for in low energy region. The black solid line is the present work. The data is from Mibe:2005er (). The units for the photon energy and are GeV.
Figure 8: Differential cross section for in the high energy region. Data are taken from Derrick:1996af (); Breitweg:1999jy ()
Figure 9: Differential cross section for . The data is from Camerini:1975cy ().

vi.4

In Figure 9 we show the differential corss section for process, and in Figure 10 we show the differential corss section for process. We note that GeV respectively, so GeV are necessary to eikonalized the heavy quarks. These results are only exploratory, since the transverse sizes of the are large in our current construction as we noted earlier. To remedy this shortcoming requires including the effects of the colored Coulomb interaction which is important in these quarkonia states. In holography this can be achieved through the use of improved holographic QCD KIRITSIS () which is beyond the scope of our current analysis.

Vii Conclusions

In QCD the diffractive photo-production of vector mesons on protons at large is described as the scattring of two fixed size dipoles running on the light cone and exchanging a soft pomeron. In a given hadron the distribution of fixed size dipoles is given by the intrinsic dipole distribution in the light cone wavefunction. The soft pomeron exchange and the intrinsic dipole distribution are non-perturbative in nature. We use the holographic construct in AdS to describe both.

The soft Pomeron parameters used in this work were previously constrained by the DIS data Stoffers:2012zw (), so the extension to the photoproduction mechanism is a further test of the holographic construction. The new parameter characterizing the transverse size of the vector mesons was adjusted to reproduce the meson radiative decays and found to be consistent with the expected string tension characteristic of the vector Regge trajectory. Comparison of our results to the data for photoproduction of vectors show fair agreement with data for the , although the inclusion of Reggeon exchanges may improve our description at low photon masses near treshold. At high photon masses, perturbative QCD scaling laws are expected. Our analysis of the photoproduction of is limited since the present construction does not account for the substantial Coulomb effects for these quarkonia. We hope to address this issue and others next.

Figure 10: Differential cross section for . The black solid line is the present work.

Viii Acknowledgements

This work was supported by the U.S. Department of Energy under Contract No. DE-FG-88ER40388. HYRyu and CHL were partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A2A01004238 and No. 2016R1A5A1013277).

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